By Guenther Hintze (auth.)
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Extra info for Fundamentals of Digital Machine Computing
SOME USEFUL THEOREMS OF BOOLEAN ALGEBRA One set of theorems is a direct extension of the postulates with the binary variable x introduced in the above equations. In this way we obtain corresponding to (3-2): x· 0=0 (3-5) x·1=x x·x=x and corresponding to (3-3): x+O=x x+ 1= 1 x+x=x (3-6) 48 DIGITAL MACHINE COMPUTING or using (3-4): x+x=1 (3-7) x·x=O All these theorems can be proven by perfect induction, substituting for x the two possible values 0 and 1. In (3-7) the primed value x is, of course, opposite to that of x.
A jump instruction expressed in the octal code 41 will be executed by signals in the binary grouping (100 001). If the content of some register in the machine is displayed on the console in binary form, the operator recognizes the octal value for each group of three binary digits. In these and many other ways the octal system is used in the communication with the computer. For computer users the familiarity with both the binary and octal system is important and calculations in the octal system are briefly explained.
3 2 1 2" 2n-1 2n-2 . . 2 2 21 2o • Binary point/ Fig. 2-4. Positional Arrangements for Integers In the above figure the notations for bit positions and bit numbers are shown side by side in order to indicate clearly the difference between the two of them. The notation of the bit position also expresses the value of the number in the register of the machine when the respective bit position contains a "1" while all other positions are "0". For example, 25 represents the 6 bit binary number (100 OOO)z = (32)10.